Dual Interpretation of the Continuity Equation and the Emergence of Cosmic Acceleration in Energy-Momentum-Squared Gravity (EMSG)

The energy-momentum squared gravity (EMSG) is a phenomenological modification of general relativity, introduced by augmenting the Einstein-Hilbert action with a quadratic term of the form $\alphaT_{\mu\nu}T^{\mu\nu}$ , where $\alpha$ is a coupling constant. Unlike many other modified gravity theories, EMSG does not naturally produce the current acceleration phase of the Universe using a pressureless (dust-like) matter component alone. As a result, to reproduce the observed cosmic acceleration, one typically supplements EMSG with either a cosmological constant or additional theoretical framework. However, this study demonstrates a novel and distinguishing feature of EMSG, dual interpretations of the continuity equation, where the second interpretation arises as a specific interaction-based reformulation of the first one. By adopting this alternative interpretation, the Friedmann equations can be redefined solely in terms of a pressureless matter fluid. The modified Friedmann equations, when constrained both theoretically and observationally, reveal that late-time cosmic acceleration can indeed emerge purely from the interaction between the Einsteinian and non-Einsteinian components of the action, without the need for any exotic energy sources such as dark energy. Consequently, EMSG, underpinned by a robust conceptual foundation and dual continuity interpretations, offers a compelling framework in which the Universe’s late-time acceleration is driven solely by matter. Remarkably, this approach allows for a unified description of both the decelerating matter-dominated era and the late-time accelerated expansion using a single matter field, ultimately reproducing the results of the ΛCDM model.